Problem 42 - Algebra 2

Question

Solve each system of equations by graphing. \(x+y=7\) \(\frac{1}{2} x-y=-1\)

Step-by-Step Solution

Answer

The solution is the point (4, 3).

Step 1: Write the Equations in Slope-Intercept Form

The slope-intercept form is given by \(y = mx + b\). For the equation \(x + y = 7\), we can rearrange it as \(y = -x + 7\). For \(\frac{1}{2}x - y = -1\), rearrange it to get \(y = \frac{1}{2}x + 1\).

Step 2: Plot the First Line

For the equation \(y = -x + 7\), identify the y-intercept as 7. Start at the point (0, 7) on the graph, then use the slope (-1) to find another point: move down 1 unit and to the right 1 unit to plot the point (1, 6). Draw a line through these points.

Step 3: Plot the Second Line

For the equation \(y = \frac{1}{2}x + 1\), the y-intercept is 1. Start at (0, 1) on the graph, then use the slope (\(\frac{1}{2}\)) to find another point: move up 1 unit and to the right 2 units to plot the point (2, 2). Draw a line through these points.

Step 4: Find the Intersection

Look where the two lines intersect on the graph. The intersection point is where both equations have the same values for \(x\) and \(y\). In this case, the lines intersect at the point (4, 3).

Step 5: Verify the Solution

Plug \(x = 4\) and \(y = 3\) back into the original equations to confirm they satisfy both. For \(x + y = 7\), \(4 + 3 = 7\). For \(\frac{1}{2}x - y = -1\), \(\frac{1}{2}(4) - 3 = 2 - 3 = -1\). Both equations are satisfied.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation of a line. It is given by the formula \(y = mx + b\). Here, \(m\) represents the slope, which indicates the steepness or inclination of the line. The \(b\) denotes the y-intercept, which is the point where the line crosses the y-axis.

To convert an equation into slope-intercept form, you need to solve for \(y\). For example, if we have an equation \(x + y = 7\), we rearrange it to \(y = -x + 7\). This tells us that the slope \(m\) is \(-1\), and the y-intercept \(b\) is \(7\).

• Remember, the slope \(m\) shows how much \(y\) changes for a change in \(x\). If \(m\) is positive, the line inclines upwards. If \(m\) is negative, it declines downwards.
• The y-intercept \(b\) is where the line will be when \(x = 0\).

Using slope-intercept form makes graphing much easier since you can easily find the starting point on the y-axis and use the slope to determine the direction and steepness of the line.

Intersection Point

The intersection point of two lines is the common point where both lines meet on the graph. This point represents the solution to a system of linear equations when graphing is used as the method for finding this solution.

In our exercise, after graphing the two lines given by the equations \(y = -x + 7\) and \(y = \frac{1}{2}x + 1\), you will find that they intersect at the point (4, 3). This means both equations are satisfied when \(x = 4\) and \(y = 3\).

• Intersection points are critical in identifying where two given conditions, represented by the equations, are true at the same time.
• It is essential to plot accurately, as any small errors can lead to incorrect conclusions about the intersection.

Once you have identified the intersection point, it’s always best to verify it by substituting back into the original equations to ensure correctness.

Solving Linear Equations

Solving linear equations is a fundamental aspect of algebra that involves finding the values of variables that satisfy given equations. When dealing with systems of equations, there are several methods to find solutions, such as graphing, substitution, and elimination. Here, we focus on solving using graphing.

By graphing each equation on a coordinate plane, we visualize where the solutions lie. Take each equation, transform it into slope-intercept form, then graph both lines to see where they intersect. This graphical representation helps in easy understanding and quick identification of the solution.

• In our problem, we use the equations \(x + y = 7\) and \(\frac{1}{2}x - y = -1\) to find the intersection point by rewriting them as lines \(y = -x + 7\) and \(y = \frac{1}{2}x + 1\), respectively.
• Always confirm the solution by replacing both \(x\) and \(y\) in the original equations. Verifying ensures your understanding and accuracy of the solution.

This technique highlights the graphical approach's simplicity in solving linear systems, making it accessible for visual learners.